An upper bound for the first nonzero Steklov eigenvalue
Li, Xiaolong Wang, Kui Wu, Haotian
Li, Xiaolong
Wang, Kui
Wu, Haotian
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2025-01-06
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Article
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Brocka-Weinstock inequality,Spherical symmetrisation,Steklov eigenvalue
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An upper bound for the first nonzero Steklov eigenvalue Xiaolong Li, Kui Wang and Haotian Wu ESAIM: COCV, 31 (2025) 5
Abstract
Let (Mn, g) be a complete simply connected n-dimensional Riemannian manifold with curvature bounds Sectg g < κ for κ < 0 and Ricg g (n - 1)Kg for K < 0. We prove that for any bounded domain Ω ⊂ Mn with diameter d and Lipschitz boundary, if Ω∗ is a geodesic ball in the simply connected space form with constant sectional curvature κ enclosing the same volume as Ω, then σ1(Ω) < Cσ1(Ωz.ast;), where σ1(Ω) and σ1(Ω∗) denote the first nonzero Steklov eigenvalues of Ω and Ω∗ respectively, and C = C(n, κ,K, d) is an explicit constant. When κ = K, we have C = 1 and recover the Brock-Weinstock inequality, asserting that geodesic balls uniquely maximize the first nonzero Steklov eigenvalue among domains of the same volume, in Euclidean space and the hyperbolic space. © The authors. Published by EDP Sciences, SMAI 2025.
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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
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EDP Sciences
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ESAIM - Control, Optimisation and Calculus of Variations
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12928119